On extremal contractions from threefolds to surfaces: the case of one non-Gorenstein point and non-singular base surface

Abstract

In this paper we study a local structure of extrmal contractions $f\colon X\to S$ from threffolds $X$ with only terminal singularities onto a surface $S$. If the surface $S$ is non-singular and $X$ has a unique non-Gorenstein point on a fiber we prove that either the linear system $|-K_X|$, $|-2K_X|$ or $|-3K_X|$ contains a "good" divisor.